Ratio prophet inequalities for convex functions of partial sums (Q1801880)

Let \(\Phi(0)\geq 0\) and assume \(\Phi\) to be convex and non-decreasing on \([0,\infty)\). Let \(X_ 1,X_ 2,\dots\) be independent mean zero random variables, \(S_ n=X_ 1+\dots+X_ n\) and \(S^*_ n=\max_{1\leq m\leq n} S^ +_ m\). Then for any \(n\geq 1\), \(E\Phi(S^*_ n)\leq 5E\Phi(S^ +_ n)\). Moreover, if the \(X\)'s are i.i.d., then the constant in this inequality can be improved to \(2-n^{-1}\leq 3\).